Open problems
Sofia Ortega Castillo
July, 2011
P
Conjecture 1 Suppose X is a Banach space such that X = ( X)p , for 1 ≤
p ≤ ∞ or p = 0, and such that X has a maximal ideal. Then an operator T
on X is a commutator if and only if T 6= λI + K, where λ 6= 0 and K is in the
maximal ideal.
Problem 2 Which operators on C(ω ω ) are commutators? Which ideals of
C(ω ω ) are closed?
Problem 3 Find the order of decay of the sequence
γ(m) =
sup
f,D,{G1m }
{||f − G1m (f, D)|||f |−1
A1 (D) },
where the supremum is taken over all dictionaries D, all elements f ∈ A1 (D) \
{0}, and all possible choices of {G1m }.
Problem 4 Denote by {an (f )} a decreasing rearrangement of absolute values
of Fourier-Walsh coefficients of f. Let a decreasing sequence {An } satisfy the
condition
X
An = o(1) as M → ∞.
M <n≤eM
Then for any f ∈ C(0, 1) with the property that an (f ) ≤ An , n = 1, 2, · · · we
have limn→∞ ||f − Gn (f )||∞ = 0. Is the converse true?
Problem 5 Is there a constant γ > 0 depending only on X and Y with the
following property: For each -isometry f : X → Y with f (0) = 0 there is a
bounded linear operator T : L(f ) → X such that
||T f (x) − x|| ≤ γ for all x ∈ X,
where L(f ) = spanf (X)?
Problem 6 For what 1 < p < ∞ does there exist f ∈ Lp (R) and Λ ⊂ R such
that {f(λ) }λ∈Λ is a basis for Lp (R) in some order?
Problem 7 Does there exist a 1-greedy basis for an infinite-dimensional complex
Banach space that is not symmetric?
Problem 8 Can Lp , 1 < p < ∞ be renormed to make (hn ) 1-greedy?
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Problem 9 Can greedy be renormed to (1 + )-greedy?
Problem 10 is the lower democracy function doubling?
Problem 11 Does the X-Greedy Algorithm converge (weakly) in Lp [0, 1] with
its usual norm?
Problem 12 Is the Metric Approximation Property equivalent to the Approximation Property for a dual space X ∗ .?
Problem 13 If X is not isomorphic to `2 , is X ergodic?
Problem 14 If X and Y are separable spaces such that X and Y are Lipschitzisomorphic, are X and Y isomorphic?
Problem 15 Can one embed an expander with geometric (T ) into a finitely
presentable group?
Problem 16 Can one give a good geometric criterion for recognizing an expander with geometric (T ) (preferable one that extends to connected graphs)?
Problem 17 Expanders with large girth are fairly well-behaved. Do they
coarsely embed into any Banach space with half-way reasonable properties(e.g.
uniformly convex, property (H) of Kasparov-Yu)?
Problem 18 Does c0 have property H?
Problem 19 Let X be an infinite dimensional Banach space. Does `2 embed
coarsely into X?
Problem 20 Is there a separable Banach space which is not Lipschitz complemented in its bidual?
Conjecture 21 There exist L, M > 0 such that for every n points in R3 one
can find at least M n2/3 of these points such that, after rotating the axes if
necessary, lie on the graph of a Lipschitz function z = f (x, y) with Lip(f ) ≤ L.
Problem 22 If X is separable, is BX an isometrically representing subset for
X?
Problem 23 If X is separable, is there K ⊂ X, K compact, such that K is an
isometrically representing subset for X?
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